In mathematics, the upper topology on a partially ordered set X is the coarsest topology in which the closure of a singleton
is a partial order, the upper topology is the least order consistent topology in which all open sets are up-sets.
However, not all up-sets must necessarily be open sets.
The lower topology induced by the preorder is defined similarly in terms of the down-sets.
Similarly, the real lower topology
is naturally defined on the lower real line
A real function on a topological space is upper semi-continuous if and only if it is lower-continuous, i.e. is continuous with respect to the lower topology on the lower-extended line
Similarly, a function into the upper real line is lower semi-continuous if and only if it is upper-continuous, i.e. is continuous with respect to the upper topology on